In the wikipedia article about the pushforward, it is stated that if $f: M\to N$ is smooth, then it induces a bundle map $df: TM \to TN$. It is then claimed that equivalently $f_*=df$ is a bundle map from $TM$ to the pullback bundle $f^* TN$.
Why is this equivalent? The bundles $TN$ and $f^* TN$ are clearly not the same as they are bundles over different spaces. They could be isomorphic, but is still seems strange that this would hold independently of $f$.
Edit: The full quote is
Equivalently (see bundle map), φ∗ = dφ is a bundle map from TM to the pullback bundle φ∗TN over M, which may in turn be viewed as a section of the vector bundle Hom(TM, φ∗TN) over M. The bundle map dφ is also denoted by Tφ and called the tangent map. In this way, T is a functor.